13. Упростите отношение: a) 15 5/6 : 8 1/3 = б) 16 2/5 : 6 5/6 = в) 5 1/6 : 6 8/9 = г) 17 3/5 : 1 31/35 =

Решение:

a) $$15 \frac{5}{6} : 8 \frac{1}{3} = \frac{15\cdot6+5}{6} : \frac{8\cdot3+1}{3} = \frac{95}{6} : \frac{25}{3} = \frac{95}{6} \cdot \frac{3}{25} = \frac{95 \cdot 3}{6 \cdot 25} = \frac{19 \cdot 5 \cdot 3}{2 \cdot 3 \cdot 5 \cdot 5} = \frac{19}{2 \cdot 5} = \frac{19}{10} = 1 \frac{9}{10}$$

б) $$16 \frac{2}{5} : 6 \frac{5}{6} = \frac{16\cdot5+2}{5} : \frac{6\cdot6+5}{6} = \frac{82}{5} : \frac{41}{6} = \frac{82}{5} \cdot \frac{6}{41} = \frac{82 \cdot 6}{5 \cdot 41} = \frac{2 \cdot 41 \cdot 6}{5 \cdot 41} = \frac{2 \cdot 6}{5} = \frac{12}{5} = 2 \frac{2}{5}$$

в) $$5 \frac{1}{6} : 6 \frac{8}{9} = \frac{5\cdot6+1}{6} : \frac{6\cdot9+8}{9} = \frac{31}{6} : \frac{62}{9} = \frac{31}{6} \cdot \frac{9}{62} = \frac{31 \cdot 9}{6 \cdot 62} = \frac{31 \cdot 3 \cdot 3}{2 \cdot 3 \cdot 2 \cdot 31} = \frac{3}{2 \cdot 2} = \frac{3}{4}$$

г) $$17 \frac{3}{5} : 1 \frac{31}{35} = \frac{17\cdot5+3}{5} : \frac{1\cdot35+31}{35} = \frac{88}{5} : \frac{66}{35} = \frac{88}{5} \cdot \frac{35}{66} = \frac{88 \cdot 35}{5 \cdot 66} = \frac{4 \cdot 22 \cdot 7 \cdot 5}{5 \cdot 3 \cdot 22} = \frac{4 \cdot 7}{3} = \frac{28}{3} = 9 \frac{1}{3}$$

Ответ: a) $$1 \frac{9}{10}$$; б) $$2 \frac{2}{5}$$; в) 3/4; г) $$9 \frac{1}{3}$$